Discretization Techniques Based on Domain Decomposition

This chapter concerns domain decomposition methods as discretization techniques for partial differential equations. We present different approaches within the framework of mortar methods [BMP93, BMP94]. Originally introduced as a domain decomposition method for the coupling of spectral elements, these techniques are used in a large class of nonconforming situations. Thus, the coupling of different physical models, discretization schemes, or non-matching triangulations along interior interfaces of the domain can be analyzed by mortar methods. These domain decomposition techniques provide a more flexible approach than standard conforming formulations. They are of special interest for time dependent problems, rotating geometries, diffusion coefficients with jumps, problems with local anisotropies, corner singularities, and when different terms dominate in different regions of the simulation domain. Very often heterogeneous problems can be decomposed into homogeneous subproblems for which efficient discretization techniques are available. To obtain a stable and optimal discretization scheme for the global problem, the information transfer and the communication between the subdomains is of crucial importance; see Fig. 1.1.